Vector-valued fragility surfaces conditioned on mainshock-aftershock intensity levels for masonry buildings

Recent earthquakes have shown the devastating impact of successive ground motions on unreinforced masonry (URM) buildings. While post-earthquake surveys provide valuable insights into the seismic performance of common construction typologies, they rarely capture damage progression induced by each event, as little or no time is available for inspection between seismic sequences. As a result, most existing vulnerability models, typically based on single mainshock (MS) analyses, underestimate the effects of cumulative damage. To address this limitation, this paper proposes a new methodology that adapts the Cloud Capacity Spectrum Method for prototype URM structures modeled using a macro-element approach based on the equivalent frame idealization. A set of physically consistent mainshock–aftershock ground motion pairs is employed to perform sequential non-linear static analyses and derive fragility surfaces directly as a function of the two events, without assuming a predefined MS-induced damage state. The resulting bivariate vulnerability surface highlights the significant amplification of expected losses due to aftershocks, even following moderate MSs. The method, demonstrated on a benchmark building and validated through a consistency check against equivalent-SDOF non-linear time history analyses, is computationally efficient and readily extendable to building portfolios.